Asymptotically isometric copies of $\ell^{1\boxplus 0}$

Year 2020, Volume: 49 Issue: 3, 984 - 997, 02.06.2020

Veysel Nezir

https://doi.org/10.15672/hujms.507488

Abstract

Using James' Distortion Theorems, researchers have inquired relations between spaces containing nice copies of $c_0$ or $\ell^1$ and the failure of the fixed point property for nonexpansive mappings especially after the fact that every classical nonreflexive Banach space contains an isometric copy of either $\ell^1$ or $c_0$. For instance, finding asymptotically isometric (ai) copies of $\ell^1$ or $c_0$ inside a Banach space reveals the space's failure of the fixed point property for nonexpansive mappings. There has been many researches done using these tools developed by James and followed by Dowling, Lennard, and Turett mainly to see if a Banach space can be renormed to have the fixed point property for nonexpansive mappings when there is failure.

In this paper, we introduce the concept of Banach spaces containing ai copies of $\ell^{1\boxplus 0}$ and give alternative methods of detecting them. We show the relations
between spaces containing these copies and the failure of the fixed point property for nonexpansive mappings. Finally, we give some remarks and examples pointing our vital result: if a Banach space contains an ai copy of $\ell^{1\boxplus 0}$, then it contains an ai copy of $\ell^1$ but the converse does not hold.

Keywords

Fixed point property, nonexpansive mapping, renorming, asymptotically isometric copy of $c_0$, asymptotically isometric copy of $\ell^1$

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